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Sum of powers

Source: Kyiv City MO 2021 Round 1, Problem 11.5

December 21, 2023
number theoryprime numbers

Problem Statement

For positive integers m,nm, n define the function fn(m)=12n+22n+32n+…+m2nf_n(m) = 1^{2n} + 2^{2n} + 3^{2n} + \ldots +m^{2n}. Prove that there are only finitely many pairs of positive integers (a,b)(a, b) such that fn(a)+fn(b)f_n(a) + f_n(b) is a prime number.
Proposed by Nazar Serdyuk
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